This phase I trial studies the side effects and best dose of autologous CD5-specific CAR-28 zeta CAR T-cells (CD5.CAR / 28 T cells) when given together with cyclophosphamide and fludarabine in treating participants with T-cell cancers expressing the CD5 antigen that that has come back. T cells, also called T lymphocytes, are special infection-fighting blood cells that can kill other cells including tumor cells. The antibody used in this study is called anti-CD5, which sticks to T-cell leukemia or lymphoma cells because of a substance on the outside of these cells called CD5. The T cells will also contain a substance called CD28 which may help stimulate the T cells and may make them last longer. Drugs used in chemotherapy, such as cyclophosphamide and fludarabine, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving CD5.CAR / 28 T cells with cyclophosphamide and fludarabine may work better in treating participants with T-cell malignancies expressing the CD5 antigen.

INCLUSION - PROCUREMENT: The first six (6) patients treated on the study should be adults (≥ 18 years [yrs] of age); six adolescents (age 12-18) should be treated before children (≥ 3 yrs of age) are eligible

INCLUSION - TREATMENT: Sexually active patients must be willing to utilize one of the more effective birth control methods during the study and for 6 months after the study is concluded; the male partner should use a condom

EXCLUSION - PROCUREMENT: History of other cancer (except non-melanoma skin cancer or in situ breast cancer or cervix cancer) unless the tumor was successfully treated with curative intent at least 2 years before trial entry

EXCLUSION - TREATMENT: Currently receiving any investigational agents or having received any tumor vaccines within the previous 6 weeks

After completion of study treatment, participants are followed up at weeks 1, 2, 3, 4, 6, and 8, and months 3, 6, 9, and 12. Then, they are followed up every 6 months for 4 years and then yearly for 10 more years.